# American Institute of Mathematical Sciences

July  2019, 39(7): 3969-4000. doi: 10.3934/dcds.2019160

## Cohomological equation and cocycle rigidity of discrete parabolic actions

 a. The MITRE Corporation, McLean, VA 22102, USA b. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Approved for Public Release; Distribution Unlimited. Case Number 18-1082. The first author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the authors. ©2018 The MITRE Corporation. All rights reserved.
1 Based on research supported by NSF grant DMS-1700837

Received  June 2018 Revised  January 2019 Published  April 2019

We study the cohomological equation for discrete horocycle maps on ${\rm SL}(2, \mathbb{R})$ and ${\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R})$ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of ${\rm SL}(2, \mathbb{R})$. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $\operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R})$, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for ${\rm SL}(2, \mathbb{R})$ in which all cases of irreducible, unitary representations of ${\rm SL}(2, \mathbb{R})$ can be studied simultaneously.

Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

Citation: James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160
##### References:

show all references

##### References:
 [1] Bassam Fayad, Raphaël Krikorian. Rigidity results for quasiperiodic SL(2, R)-cocycles. Journal of Modern Dynamics, 2009, 3 (4) : 479-510. doi: 10.3934/jmd.2009.3.479 [2] Artur Avila, Thomas Roblin. Uniform exponential growth for some SL(2, R) matrix products. Journal of Modern Dynamics, 2009, 3 (4) : 549-554. doi: 10.3934/jmd.2009.3.549 [3] Artur Avila. Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension. Journal of Modern Dynamics, 2009, 3 (4) : 631-636. doi: 10.3934/jmd.2009.3.631 [4] Russell Johnson, Mahesh G. Nerurkar. On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows. Communications on Pure & Applied Analysis, 2011, 10 (3) : 873-884. doi: 10.3934/cpaa.2011.10.873 [5] Ser Peow Tan, Yan Loi Wong and Ying Zhang. The SL(2, C) character variety of a one-holed torus. Electronic Research Announcements, 2005, 11: 103-110. [6] Julie Déserti. Jonquières maps and $SL(2;\mathbb{C})$-cocycles. Journal of Modern Dynamics, 2016, 10: 23-32. doi: 10.3934/jmd.2016.10.23 [7] Samuel C. Edwards. On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$). Journal of Modern Dynamics, 2017, 11: 155-188. doi: 10.3934/jmd.2017008 [8] Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120. [9] Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $\mathbb{R}^2$ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125 [10] Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985 [11] Anatole Katok, Federico Rodriguez Hertz. Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 609-615. doi: 10.3934/dcds.2010.27.609 [12] Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $\mathbb{R}^2$ with nonlinearities having new exponential growth condition. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211 [13] Guji Tian, Qi Wang, Chao-Jiang Xu. $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1023-1039. doi: 10.3934/dcds.2016.36.1023 [14] Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012 [15] J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665 [16] Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725 [17] Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088 [18] Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030 [19] Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks & Heterogeneous Media, 2012, 7 (4) : 837-855. doi: 10.3934/nhm.2012.7.837 [20] Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2703-2728. doi: 10.3934/dcds.2014.34.2703

2019 Impact Factor: 1.338